System and method for shaping ultra wide bandwidth signal spectrum

ABSTRACT

A method and system shapes a spectrum of an impulse radio signal, such as an UWB signal. First, basis pulses at various frequencies and pseudo-random delays are generated. The generated set of basis pulses are then weighted and delayed, and combined linearly to conform the spectrum of the transmitted basis pulses to a spectral mask. The spectral mask can be predetermined, or the conforming can be adaptive. Furthermore, the basis pulses can be frequency-shifted before the combining.

CROSS REFERENCE TO RELATED APPLICATION

A claim of priority is made to U.S. provisional application Ser. No.60/451,577, “UWB Communication system with shaped signal spectrum bypulse combination,” filed Mar. 3, 2003.

FIELD OF THE INVENTION

This invention relates generally to wireless communications, and moreparticularly to shaping the spectrum of transmitted signals in ultrawide bandwidth communication systems.

BACKGROUND OF THE INVENTION

With the release of the “First Report and Order,” Feb. 14, 2002, by theFederal Communications Commission (FCC), interest in ultra widebandwidth (UWB) communication systems has increased. The IEEE 802.15standards organization, which is responsible for personal area networks(PAN), has established a task group, TG3a, to standardize ahigh-data-rate physical layer based on UWB. One of the most importantrequirements is compliance with the FCC spectral mask for indoorchannels shown in FIG. 1. The spectral mask shows power limits as afunction of frequency.

In addition to this requirement, it is also required by IEEE 802.15 thatUWB systems do not interfere with existing wireless systems, such as802.15.1 (Bluetooth), 802.15.3 (personal area networks), 802.15.4(Zigbee) and 802.1 a and 802.11b (wireless local area networks).

Furthermore, UWB systems should also be resilient against interferencefrom these existing wireless systems. In addition, UWB systems shouldalso resist interference from home appliances, such as microwave ovens,and other narrowband interferers.

All these requirements impose additional constraints on the spectrumshaping. Minimizing interference, in both receivers and transmitters,requires placing minima on the poser in the transmission spectrum.Rigorously speaking, the spectral mask demands very low power in certainfrequency bands, as well as matching filters. A null in the transmitspectrum also implies a null in the transfer function of the matchingfilter, and thus interference suppression at this frequency.

While some interference, e.g., microwave radiation, is at fixedfrequencies, other interference is at variable frequencies, e.g., thedifferent bands of 802.11a, or at frequencies that cannot be predicted apriori. Thus, it is necessary to be able to shape the spectrumadaptively.

For impulse radio signals, pulse-position modulation, (PPM) and pulseamplitude modulation (PAM) are the most popular signaling techniques.These techniques are combined with time hopping (TH) for multiple-accessformat. In TH, each symbol is represented by a series of “basis pulses.”The location or delay of the pulses is determined by a time-hoppingcode. The time-hopping code is generated as a pseudo-random sequence.

In general, the spectrum with random code generation can be quitedifferent from the FCC mask. As a result, in order to be compliant withFCC regulation, with a scaling of transmission power, the maximumallowed transmission power can be significantly restricted. Therefore,any signal shaping method that uses a basic, single short pulse mustexplicitly take the FCC mask into consideration.

Under certain conditions the spectrum of the transmit signal becomesidentical to the spectrum of the basis pulse. Therefore, there is a needto modify the spectrum of the basis pulse to fit the requirements ofboth the FCC mask and industry standards.

SUMMARY OF THE INVENTION

The invention shapes the spectrum of a transmitted signal in an ultrawide bandwidth (UWB) system so that the transmitted signal meetsregulatory constraints, as well as practical requirements ofinterference suppression.

The system according to the invention uses a linear combination ofweighted, time-shifted (delayed) basis pulses in order to achieve adesired spectrum. Characteristically, the system uses a bank of basicpulses, preferably with different time-frequency duration, toapproximate a given mask of general shape with a small number ofparameters.

The weights and delays are selected according to criteria related tospectral efficiency, adherence to the FCC spectral mask, and otherrestrictions like maximum number of rake fingers.

The method for determining the weights is based on a non-linear search,initialized by the solution of a quadratic optimization problem. Twoways of approximating the original problem with a quadratic formulationare described. Additionally, the invention provides combinatorialoptimization to perform discrete optimization over pulse positions(delays), as well as off-line selection from a basic pulse set.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph of a spectral mask used by the invention;

FIG. 2 is a block diagram of a system for shaping an ultra widebandwidth spectrum according to the invention;

FIG. 3 is a block diagram of the system of FIG. 1 further including aset of oscillators according to the invention;

FIG. 4 is a block diagram of a feed-forward neural network used by theinvention; and

FIG. 5 is a block diagram of feed-backward neural network for adaptiveoptimization with back-propagation according to the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

As shown in FIG. 2, our invention uses a linear combination 110 of a setof basis pulses 101 for shaping of a spectrum a transmitted impulseradio signal, s(t) 102, for example, a transmitted ultra wide bandwidth(UWB) signal. The basis pulses 101 are generated using circuits so thatthey are spread over the frequency spectrum with pseudo random delays.

Prior to the combining 110, a set of optimized filter and delay lines120, h_(l)(t), l=1, . . . , L, is applied to the pulses to weight anddelay the generated basis pulses to conform to a predetermine spectralmask.

The combination of the filtered pulses can be achieved by a combinationof analogue delay lines, adders, and programmable pulse generators.

As shown in FIG. 3, our invention can also apply a set of oscillators310 to the output of the filters 120 to shift the frequency, ifnecessary. The oscillators are of the form cos(Ω_(i)t+Φ_(i)). Frequencyshifting enables an additional degree of freedom in the design, otherthan the delaying and weighting as shown in FIG. 2.

An important part of our method is the determination of the pulseweights and delays. We determine the weight and delays on either fixedor time-varying criteria for the spectrum shape of the resulting signal102. These criteria can stem from the FCC spectral mask (fixed), fromthe necessity to avoid interference to other users, which can bepre-defined or time-varying, or following an instantaneous or averageddetermination of the emissions of users in the current environment, orother criteria. In any case, these criteria are mapped onto an“instantaneous’ spectral mask that has to be satisfied by the resultingsignal 102.

Selecting Pulses Off-Line

The selection of the basis pulses 101 can be based on two criteria:implementation complexity, and approximation capability.

We prefer the basis pulses to be Gaussian in form. Gaussian pulses arerelatively easy to generate, and their differentiations and integrationscan be implemented with differentiation and integral circuits well knownin the art.

The approximation can be performed with multi-resolution time-frequencyanalysis of signals using, for example, wavelets, and multi-resolutiontime-frequency approximation.

The signal of interest for the invention conforms to a desired spectralmask. Whereas a signal may be analyzed with respect an infinite numberof possible bases in signal space, the choice of bases matterscritically when one is restricted to a given number of bases availableto approximate the spectrum of the desired signal.

A high-level description of the problem is given as follows. Assumethere is a set of candidate of basis functions C. For example, the basisfunctions correspond to differentiations of a Gaussian pulse of variousorders. In addition, the basis functions can be orthogonalized, e.g., bya sequential Gram-Schmidt procedure, to facilitate the subsequentoptimization.

A set of typical masks available for training purpose is denoted byS={M(Ω)}, and one particular set of chosen bases as denoted by p(t). Theparticular set is obtained by stacking the selected basis functions intoa column vector.

We define a function f (p(t), S) to evaluate the fulfillment of the twoconsiderations. For example, the function is a weighted combination oftwo parts: the first part models a cost of generating the pulse, and thesecond part models an approximation error or some other efficiencymetric. This function can be expressed as the formulation:${{f\left( {{\underset{\_}{p}(t)},S} \right)} = {{\alpha\quad{f_{1}\left( {\underset{\_}{p}(t)} \right)}} + {\beta{\sum\limits_{{M{(\Omega)}} \in S}\quad{f_{2}\left( {{\underset{\_}{p}(t)},{M(\Omega)}} \right)}}}}},$where α and β are predetermined constants. This formulation is acombinatorial optimization problem, described below.

In comparison, the problem of optimal pulse locations is mainly foron-line applications, where the choices of bases are fewer, theimplementation cost is fixed, and there is only one target mask, insteadof a set of training pulses.

Formulation of the Problem

The individual basic pulses are denoted by p_(l)(t), l=1 . . . L, andtheir Fourier transforms by P_(l)(jΩ), l=1, . . . L. The set of shapingfilter 120 is FIR, with the impulse response being the sum ofδ-functions placed at different delays, τ, and weighted or scaled, s,differently. Corresponding to FIG. 2, we have $\begin{matrix}{{{h_{l}(t)} = {\sum\limits_{i = 0}^{M_{l}}\quad{s_{li}{\delta\left( {t - \tau_{li}} \right)}}}},} \\{{{s(t)} = {\sum\limits_{l = 1}^{L}\quad{\sum\limits_{i = 0}^{M_{l}}\quad{s_{li}{p_{l}\left( {t - \tau_{li}} \right)}}}}},} \\{{S\left( {j\quad\Omega} \right)} = {\int_{- \infty}^{\infty}{{s(t)}{\mathbb{e}}^{- {j\Omega t}}\quad{\mathbb{d}t}}}} \\{= {\sum\limits_{l = 1}^{L}\quad{\sum\limits_{i = 0}^{M_{l}}\quad{s_{li}{P_{l}\left( {j\quad\Omega} \right)}{{\mathbb{e}}^{- {j\Omega\tau}_{li}}.}}}}}\end{matrix}$We use the following notations:${\underset{\_}{s} \equiv \left\lbrack {s_{10}\quad\ldots\quad s_{1M_{1}}\quad\ldots\quad s_{L\quad 0}\quad\ldots\quad s_{{LM}_{L}}} \right\rbrack^{T}},{{\underset{\_}{p}(t)} \equiv \left\lbrack {{{p_{1}\left( {t - \tau_{10}} \right)}\quad\ldots\quad{p_{1}\left( {t - \tau_{1M_{1}}} \right)}\quad\ldots\quad{p_{L}\left( {t - \tau_{L\quad 0}} \right)}\quad\ldots\quad{p_{L}\left( {t - \tau_{{LM}_{L})}} \right\rbrack}^{T}},{{\underset{\_}{P}\left( {j\quad\Omega} \right)} \equiv {\left\lbrack {{P_{1}({j\Omega})}{\mathbb{e}}^{- {j\Omega\tau}_{10}}\quad\ldots\quad{P_{1}\left( {j\quad\Omega} \right)}{\mathbb{e}}^{- {j\Omega\tau}_{1M_{1}}}\quad\ldots{P_{L}({j\Omega})}{\mathbb{e}}^{- {j\Omega\tau}_{L\quad 0}}\quad\ldots\quad{P_{L}\left( {j\quad\Omega} \right)}{\mathbb{e}}^{- {j\Omega\tau}_{{LM}_{L}}}} \right\rbrack^{T}R} \equiv {\int_{- \infty}^{- \infty}{{\underset{\_}{p}(t)}{\underset{\_}{p}(t)}^{T}\quad{\mathbb{d}t}}}},{{\left\langle {{s(t)},{s(t)}} \right\rangle \equiv {\int_{- \infty}^{- \infty}{{s(t)}^{2}{\mathbb{d}t}}}} = {{\underset{\_}{s}}^{T}R{\underset{\_}{s}.}}}} \right.}$

The elements of p(t) constitute a pool of bases pulses in the signalspace. Then, the single user spectrum shaping problem can now beformulated as follows:${\max\limits_{\underset{\_}{s}}\left\langle {{s(t)},{s(t)}} \right\rangle},{{{subject}\quad{to}\quad{{S\left( {j\quad\Omega} \right)}}^{2}} < {M(\Omega)}},{\forall{\Omega \in \left\lbrack {{- \infty},{+ \infty}} \right\rbrack}},$where M(Ω) is an upper-bound on the squared magnitude response regulatedby FCC.

This is equivalent to a min-max formulation:${\min\limits_{\underset{\_}{s}}{\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}\frac{{{S({j\Omega})}}^{2}}{M(\Omega)}}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}R\underset{\_}{s}} = 1.}$

In addition, structural constraints can be put on the parameter vector sdue to implementation concerns. For example, the number of non-zeroelements in certain sub-vectors of the parameter vector s can beconstrained.

In certain scenarios, the inner product of the desired signal <s(t),s(t)> can be replaced by the maximum of the signal energy within acertain frequency range, e.g., the “in-band” proportions of energy aremaximized. This only changes the definition of the matrix R.

In certain scenarios, the constraints |S(jΩ)|²<M(Ω), ∀Ω∈=[−∞,+∞] areextended to include integral spectrum constraints, e.g., for“out-of-band” signals, which produces interference to other devices.These constraints are not as strict as the fixed envelop definition ofthe FCC mask.

The modifications corresponding to the system shown in FIG. 3 followafter noting: $\begin{matrix}{{\int_{- \infty}^{- \infty}{{\mathbb{e}}^{- {j\Omega t}}{\cos\left( {{\Omega_{l}t} + \varphi_{l}} \right)}{p_{l}\left( {t - \tau_{li}} \right)}\quad{\mathbb{d}t}}} = {\frac{1}{2}{P_{l}\left( {j\left( {\Omega - \Omega_{l}} \right)} \right)}}} \\{{{\mathbb{e}}^{- {j{({\Omega - \Omega_{l}})}}}{\mathbb{e}}^{{j\varphi}_{l}}} +} \\{\frac{1}{2}{P_{l}\left( {j\left( {\Omega + \Omega_{l}} \right)} \right)}} \\{{\mathbb{e}}^{- {j{({\Omega + \Omega_{l}})}}}{\mathbb{e}}^{- {j\varphi}_{l}}}\end{matrix}$

Initializing Pulse Parameters

The min-max formulation as described above, or a robust ∞-normminimization is known to be a difficult problem. Several existinggame-theoretic techniques rely on the existence of saddle points, whichunfortunately are not satisfied in this case.

Therefore, we minimize with an approximate 2-norm formulation instead.In other words, we replace the maximum over all frequency by the minimumof a weighted integral formulation, i.e.,${\min\limits_{\underset{\_}{s}}{\int_{- \infty}^{+ \infty}{{w(\Omega)}\frac{{{S({j\Omega})}}^{2}}{M(\Omega)}{\mathbb{d}\Omega}}}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}R\underset{\_}{s}} = 1.}$${{We}\quad{also}\quad{note}\quad{{that}:{\int_{- \infty}^{+ \infty}{{w(\Omega)}\frac{{{S({j\Omega})}}^{2}}{M(\Omega)}{\mathbb{d}\Omega}}}}} = {{\int_{- \infty}^{+ \infty}{{w(\Omega)}\frac{{\underset{\_}{s}}^{T}{\underset{\_}{P}({j\Omega})}{\underset{\_}{P}({j\Omega})}^{H}\underset{\_}{s}}{M(\Omega)}{\mathbb{d}\Omega}}} = {{{{\underset{\_}{s}}^{T}\left( {\int_{- \infty}^{+ \infty}{{w(\Omega)}\frac{{\underset{\_}{P}({j\Omega})}{\underset{\_}{P}({j\Omega})}^{H}}{M(\Omega)}{\mathbb{d}\Omega}}} \right)}\underset{\_}{s}} \equiv {{\underset{\_}{s}}^{T}W{\underset{\_}{s}.}}}}$

Provided that there is no structural constraint on s, the quadraticapproximation leads to:${\min\limits_{\underset{\_}{s}}{{\underset{\_}{s}}^{T}W\underset{\_}{s}}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}R\underset{\_}{s}} = 1},$for which the solution is an eigenvector corresponding to the smallesteigenvalue λ of the matrix W−λR. When there are structural constraintson s and the non-zero positions have been fixed, e.g., as one elementarystep in a combinatorial optimization procedure, the problem bears asimilar form except that W and R are now replaced by their correspondingprincipal sub-matrices.

Optimization Over Pulse Positions

One notable difference of our invention, compared with traditionalleast-squares FIR filter design, is that the pulse positions, i.e.,delays of the non-zero elements in s are also optimized.

In prior art UWB systems, the random pulse-positive modulation reliesessentially on a sparse arrangement of the pulses to supportasynchronous transmissions, to provide better multi-path resolution, andto avoid catastrophic collisions.

The optimization over the pulse positions or delays is a combinatorialoptimization problem, which determines an optimal subset selected from agiven set of basis functions p(t).

As stated above, this problem is in similar form as the optimal choiceof the initial basis functions, i.e., p_(l)(t−τ_(li)), from a larger setof candidate pulses. The second cost function f₂(p(t), M(Ω)) can beselected as an extreme value of the min-max, or the approximatemin-integral formulation.

Here, we describe three selection procedures, with the first two beinggreedy.

Iterative Greedy Addition

We initialize with an empty set of basis functions.

We search for a basis function from the candidate pool, which maximizesthe cost function when added to the existing set of basis functions.

We select this basis function, and we repeat the search until thedesired number of basis functions has been selected.

Iterative Greedy Removal

We initialize with the entire set of basis functions.

We search for a basis function from the candidate pool, which maximizesthe cost function when removed from the existing set of basis functions.

We select this basis function, and we repeat the search until thedesired number of basis functions has been selected.

Branch-and-Bound

This approach assumes that p_(l)(t−τ_(li)) have already beenorthogonalized and normalized. Additionally, this approach uses aquadratic approximation rather than the min-max formulation. In thiscase, R=I, and the minimum generalized eigenvalue of matrix pair (W, R)reduces to the minimum eigenvalue of W, which is the cost function to beminimized.

Let K denote the maximum number of non-zero elements. Let N denote thenumber of candidate pulses. Denote one arrangement of the pulsepositions (delays) by a vector with K 1's, e.g., (1, 0, 0, 1, . . . ,0).

In the following, we construct an enumeration tree of all the allowedcombinations, where each node n is associated with a positioning vectorwith each element from the set {0, 1, *} where * means the position maybe a 1 or 0. The enumeration tree is defined as follows:

The root of the tree is (*,*, . . . ,*), referring to all possibleselections.

The first layer of the tree enumerates all the vectors with one 0 andN−1*'s by considering the placement of a 0.

The children of any node with less than N−K zeros enumerates all thepossibilities of putting another 0. To avoid duplications, ( . . . ,0,*, . . . ,*) is by convention associated with one possible father node( . . . ,*,*, . . . ,*), where the first dotted part is the same as thecurrent child vector.

At each visited node, the procedure maintains a lower bound and an upperbound of the cost function among all the descendant combinations. Theprocedure visits the enumeration tree with certain order, e.g.,depth-first, or breadth-first. When visiting a certain node, theprocedure updates the upper-bound and lower-bound of the cost function.If the lower-bound is greater than the minimum upper-bound among thevisited nodes, the sub-tree, with its root being the current node, ispruned from further consideration. The bounds are determined by Cauchy'sinterlacing theorem of eigenvalues for symmetric matrices:

Cauchy's Interlacing Theorem:

Let Ar denote a r×r submatrix of a n×n real symmetric matrix A. Theeigenvalues λ₁≦λ₂≦ . . . ≦λ_(n) of A, and the eigenvalues μ₁≦μ₂≦ . . .≦μ_(r) of A_(r) satisfy the following relations: λ_(i)≦μ_(i)≦μ_(i+n−r),i=1, . . . ,r.

The procedure can terminate after visiting all the nodes. In this case,the procedure reaches a global optimal selection of pulse delays.Alternatively, the procedure can terminate when the cost functionevaluated at a visited node is within ε of the current upper-bound ofall the possible combinations.

Quadratic Optimization for L=1

For the special case of L=1 and τ_(li)=iΔτ, i=1, . . . M₁, a distinctquadratic approximation is pursued. In this formulation, theoptimization over the pulse positions is directly solvable.

We introduce${{G(\Omega)} = \frac{\sqrt{M(\Omega)}}{{P({j\Omega})}}},{{\phi\left( {\Omega,\underset{\_}{s}} \right)} \equiv {{{s_{10} + {s_{11}{\mathbb{e}}^{- {j\Omega\Delta\tau}}} + \ldots + {s_{1M_{1}}{\mathbb{e}}^{{- {j\Omega M}_{l}}{\Delta\tau}}}}}^{2}.{Then}}},{{\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}\frac{{{S({j\Omega})}}^{2}}{M(\Omega)}} = {\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}{\frac{\phi\left( {\Omega,\underset{\_}{s}} \right)}{{G(\Omega)}^{2}}.{Note}}}},{{{\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}\frac{\sqrt{\phi\left( {\Omega.\underset{\_}{s}} \right)}}{G(\Omega)}} - 1} = {\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}{\frac{\sqrt{\phi\left( {\Omega,\underset{\_}{s}} \right)} - {G(\Omega)}}{{G(\Omega)}^{2}}.}}}$

Now, we select a substantially large constraint, i.e., s ^(T)Rs=b, suchthat the above equation is equal to the ∞-norm. Although it does notmake any difference in the original ∞-norm formulation, it does affectthe 2-norm approximation. With the 2-norm approximation, the problembecomes a FIR filter design problem with a least-squares formulation:$\min\limits_{\underset{\_}{s}}{\int_{\Omega}{{{s_{10} + {s_{11}{\mathbb{e}}^{- {j\Omega\Delta\tau}}} + \ldots + {s_{1M_{1}}{\mathbb{e}}^{{- {j\Omega M}_{l}}{\Delta\tau}}{{- {G(\Omega)}}}^{2}\quad{\mathbb{d}\Omega}}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}R\underset{\_}{s}} = {b\quad{with}\quad\left. b\uparrow\infty \right.}}}}}$

This equation can be approximated by${\min\limits_{\underset{\_}{s},n}{\sum\limits_{i = 0}^{M_{1}}\quad\left( {s_{i} - g_{i + n}} \right)^{2}}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}R\underset{\_}{s}} = {b\quad{with}\quad\left. b\uparrow\infty \right.}},$where g_(n) is a real sequence that produces an amplitude response${G\left( {\mathbb{e}}^{j\quad w} \right)} \equiv {{G\left( \frac{w}{\Delta\quad\tau} \right)}.}$

In particular, we match the non-zero positions to the taps of g_(n) withlargest magnitudes. After, the positions of the taps have been fixed,the solutions to the tap weights are computed as s∝u ₁ u ₁ ^(T) g, whereu_(n) is the principal component of R, and g are the vectors constructedfrom elements of g_(n) having largest magnitudes among those complyingwith the structural constraints of s.

Note, the above two quadratic approximations are only two out of themany possible quadratic approximations. In essence, without FIRconstraints and the degree of freedom constraints, the optimal solutionis an IIR filter, which uniformly matches the FCC spectral mask.

The various quadratic approximations can be viewed as minimizing somedistance measure between the pursued FIR solution and the IIR solution.We prefer the above quadratic approximation mainly because the jointoptimizations over the weights and delays are easily solvable due to itsspecial structure.

Non-linear Optimization with Neural Networks

FIG. 4 shows a feed-forward network with a differential soft-maximum,410, and FIR frequency-responses at irregularly spaced frequencies 420to evaluate the function. In FIG. 4, the weighting or scalingcoefficients are s 401, the pulse positions or delays are τ 402, Φ_(i)represents |φ_(i)(Ω_(i),s,τ)|², e^(ax) 431 represents the exponentialoperator, and Ψ411 represents the output of the soft maximization, i.e.,the sum of the exponentials 431.

We initialize our solution with the above quadratic approximation, andfurther exploit non-linear optimization techniques to gradually refinethe solutions. The description in this section refers to the originalmin-max formulation described above.

A back-propagated (BP) multi-layer perceptron (MLP) possesses adaptivelearning abilities to estimate sampled functions, represent thesesamples, encode structural knowledge, and inference inputs to outputsvia association. Its main strength lies in its substantially largenumber of hidden units, and thus, a large number of interconnections.The MLP neural networks enhance the ability to learn and generalize fromtraining data.

We describe the MLP optimization for the special case L=1. Optimizationof the scaling coefficients, with fixed positions in the general caseL>1, follows similarly. Because L=1, all quantities that depends on Lare simplified to shortened notations with the dependence removed.

In addition, we re-parameterize the problem so that s₁, . . . , s_(k)represent the scaling or weighting coefficients at the pulse locations(delays) τ₁, . . . τ_(K). Therefore, we define τ≡[τ₁, . . . τ_(K)]^(T).The optimization over pulse positions for L=1 can be treated by MLP, forL>1. We refer again to the three combinatorial optimization approachesdiscussed earlier.

In order to put the current problem into the general framework of MLP,we uniformly quantize the frequency range to arrive at${\max\limits_{\Omega \in {\lbrack{{- \infty},{+ \infty}}\rbrack}}\frac{{{S\left( {j\quad\Omega} \right)}}^{2}}{M(\Omega)}} \approx {\max\limits_{i \in {\{{0,1,{{\ldots\quad N} - 1}}\}}}{\frac{{S\left( {j\quad\Omega_{i}} \right.}^{2}}{M\left( \Omega_{i} \right)}.{rthrh}}}$Then, we replace the max-function with a differentiable soft-max. Givena function f(x)>0, x∈S, for a sufficiently large positive number α, wehave the following soft-max approximation:${\underset{x \in S}{\max\quad}{f(x)}} \approx {\frac{1}{\alpha}\ln{\sum\limits_{x \in S}^{\quad}\quad{{\mathbb{e}}^{\alpha\quad{f{(x)}}}.}}}$Then, we replace the max-function with a differentiable soft-max. Givena function f(x)>0,x∈S, for a sufficiently large positive number α, wehave the following soft-max approximation:${\underset{x \in S}{\max\quad}{f(x)}} \approx {\frac{1}{\alpha}\ln{\sum\limits_{x \in S}^{\quad}\quad{{\mathbb{e}}^{\alpha\quad{f{(x)}}}.}}}$

With these two simplifications, the problem becomes${\min\limits_{\underset{\_}{s},\underset{\_}{\tau}}{\psi\left( {\underset{\_}{s},\underset{\_}{\tau}} \right)}},{{{subject}\quad{to}\quad{\underset{\_}{s}}^{T}{R\left( \underset{\_}{\tau} \right)}\underset{\_}{s}} = 1},$with R(τ) being the principal submatrix of R corresponding to delays τand${{\psi\left( {\underset{\_}{s},\underset{\_}{\tau}} \right)} \equiv {\sum\limits_{i = 0}^{N - 1}\quad{\mathbb{e}}^{\alpha\quad{\phi_{i}{({\underset{\_}{s},\underset{\_}{\tau}})}}{G{(\Omega_{i})}}}}},{{\phi_{i}\left( {\underset{\_}{s},\underset{\_}{\tau}} \right)} \equiv {{\phi\left( {\Omega_{i},\underset{\_}{s},\underset{\_}{\tau}} \right)}.}}$

FIG. 5 shows the feed-backward network 500 for adaptive optimizationwith back-propagation, where 501 and 502 are the derivatives defined bythe equations in FIG. 5.

While it is theoretically possible to simultaneously adjust both thepositions or delays τ 401 and the weighting or scaling coefficients s402, practically we prefer to decouple their tuning by adopting aconditional maximization approach, i.e., optimizing one with the otherfixed. In addition, this decoupling may be justified by the differentnature of the two parameter sets. In preferred embodiment, with theallowed positions quantized, the slight change τ→τ+μδ_(tτ) is alwaysnormalized to “hop” to the nearest valid quantization point on themulti-dimensional grid.

Typically, numerical non-linear optimizations can only be assured toarrive at a local optima rather than a global one. Simulated annealingcan be used to avoid local optima. One approach to escape from possiblelocal minima is to begin the optimization with several randomlydistributed initial solutions, and select the best solution among thedifferent trial paths.

We note that this is only one possible neural network, which again isjust one of the methods for implementing non-linear optimization.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications may be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

1. A method for shaping a spectrum of an impulse radio signal,comprising: generating a set of ultrawide bandwidth basis pulses at aplurality of frequencies and a plurality of random delays; optimizing,jointly, weights and delays as a solution to a quadratic optimizationproblem to approximately minimize a deviation of the spectrum from anultrawide bandwidth spectral mask, in which the spectral mask isdesigned for indoor channels and limits power as a function of frequencyin the spectral mask; orthogonalizing and normalizing the set ofultrawide bandwidth basis pulses; and applying a branch and boundprocedure to the set of orthogonalized and normalized ultrawidebandwidth basis pulses to optimize the delays. weighting the set ofultrawide bandwidth basis pulses by the weights; delaying the set ofbasis pulses by the random delays; and combining linearly the weightedand delayed basis pulses to conform the spectrum to the ultrawidebandwidth spectral mask, and wherein the weights and delays are fixedover time for the spectral mask, and wherein the ultrawide bandwidthbasis pulses are selected from a set of basis pulses by a combinatorialoptimization using training spectral masks.
 2. A method of claim 1further comprising: shifting frequencies of the weighted and randomlydelayed ultrawide bandwidth basis pulses before the combining. 3.(canceled)
 4. The method of claim 1 wherein the weights and delays varyover time to adaptively shape the spectrum.
 5. The method of claim 1wherein the ultrawide bandwidth basis pulses are Gaussian in form. 6.The method of claim 1 wherein the weighting and delaying are performedby a set of filters and a set of delay lines, and the combining isperformed by an adder.
 7. The method of claim 1 further comprising:evaluating a cost function to determine the weights and delays.
 8. Themethod of claim 7 wherein the cost function,f, includes first and secondfunctions f₁ and f₂, and${{f\left( {{\underset{\_}{p}(t)},S} \right)} = {{\alpha\quad{f_{1}\left( {\underset{\_}{p}(t)} \right)}} + {\beta{\sum\limits_{{M{(\Omega)}} \in S}^{\quad}\quad{f_{2}\left( {{\underset{\_}{p}(t)},{M(\Omega)}} \right)}}}}},$where α and β are predetermined constants, S=^(M(Ω)) denote the spectralmask, and p(t) denotes the set of basis pulses, and the first functionf₁ models a cost of generating the basis pulses, and the second functionf₂ models an approximation error.
 9. The method of claim 1 wherein thedelays are fixed, and further comprising: solving a quadraticoptimization problem to approximately minimize a deviation from thespectral mask.
 10. The method of claim 9 further comprising: refiningthe weights and delays by a non-linear optimization.
 11. The method ofclaim 10 wherein the non-linear optimization is performed by aback-propagation neural network.
 12. The method of claim 10 wherein thenon-linear optimization is performed by a multiple-layer neural network13. The method of claim 10 wherein the non-linear optimization isperformed by a simulated annealing process.
 14. (canceled)
 15. Themethod of claim 1 further comprising: selecting the set of basis pulsesfrom a candidate set of basic pulses by greedy addition to optimize thedelays.
 16. The method of claim 1 further comprising: selecting the setof basis pulses from a candidate set of basic pulses by greedy removalto optimize the delays.
 17. The method of claim 1 further comprising:orthogonalizing and normalizing the set of basis pulses; and applying abranch and bound procedure to the set of orthogonalized and normalizedbasis pulses to optimize the delays.
 18. The method of claim 1 whereinbounds of the branch and bound procedure are determined by Cauchy'sinterlacing theorem of eigenvalues for symmetry matrices.
 19. The methodof claim 1 wherein the branch and bound procedure further comprises:constructing an enumeration tree with an increasing number of zeros invectors representing the delays.
 20. (canceled)
 21. A system for shapinga spectrum of an impulse radio signal, comprising: means for generatinga set of ultrawide bandwidth basis pulses at a plurality of frequenciesand a plurality of random delays means for optimizing, jointly, weightsand delays as a solution to a quadratic optimization problem toapproximately minimize a deviation of the spectrum from an ultrawidebandwidth spectral mask, in which the spectral mask is designed forindoor channels and limits power a function of frequency in the spectralmask; a set of delay lines configured to delay the set of basis pulsesby the random delays; and an adder configured to combine linearly theweighted and delayed basis pulses to conform the spectrum to theultrawide bandwidth spectral mask, and wherein the ultrawide bandwidthbasis pulses are selected from a set of basis pulses by a combinatorialoptimization using training spectral masks.
 22. The system of claim 21further comprising: a set of oscillators configured to shift frequenciesof the weighted and delayed basis pulses before the combining.